For the arc with radius \( a \) and angle \( \frac{3\pi}{2} \), the magnetic field at the origin is: \[ B_1 = \frac{\mu_0 I}{4\pi a} \]
Step 2: Magnetic Field Due to the Straight SegmentFor the straight segment of the wire: \[ B_2 = \frac{\mu_0 I}{4\pi a} \left( \frac{3\pi}{2} \right) \]
Step 3: Magnetic Field Due to Other SegmentsSince the magnetic field due to the straight segments at the origin is zero: \[ B_3 = 0 \]
Step 4: Calculate the Total Magnetic FieldThus, the total magnetic field at the origin is: \[ B = \frac{\mu_0 I}{4\pi a} \left( \frac{3\pi}{2} \right) \]
Match List - I with List - II.